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CAARMS11 Tutorial on Lagrangians

28 pages
11th Conference for African American Researchers in the Mathematical Sciences Institute for Pure and Applied Mathematics, June 21-24, 2005 Tutorial on the Calculus of Variations:Part I, LagrangiansWilliam A Massey,Princeton Universitywmassey@princeton.edu1CAARMS ACKNOWLEDGEMENTSThanks to Prof. Rudy L. Horne for pointing out to me the connections between optimization and Hamiltonian systems.These notes also benefited from many discussions withProf. Earl Barnes.2””Dot Notation and Other Simplifications• dx(t) x(t)dt• •L x,x L x(t),x(t),t( ) ( )3£”£Action Integrals and LagrangiansOur goal is to find a differentiable curve of the formx x(t) 0 t T that is extremal for (maximizes or { }minimizes) the action integral as given belowT• •A(x,x,T) = L x(t),x(t),t dt( )∫0The integrand function L is called the Lagrangian and is assumed to be continuously differentiable in all its arguments. 4”££Application Interpretation for Action Integrals and Lagrangians:Operations ResearchThe curve x x(t) 0 t T is now the evolution of{ }the state of some system and the Lagrangian L is theprofit (or cost) rate function. The action integral is the total profit (cost) over [0,T]. We use calculus of variations to maximize the profit (or minimize the cost) over [0,T].5TELEPHONE CALL CENTERS6Call Centers Mechanicsll ntrhicNumber of Nuber of PositionPositionCustomers in Systemustoers in SysteCustomer Flow Rate Velocityustoer Flo ate ...
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